Why the 5-to-10-year forward beats 20 percent

The annually compounded five-year rate is $10\%$ and the ten-year rate is $15\%$. The quick estimate for the forward rate from year 5 to year 10 is $2\times15\%-10\%=20\%$. Explain in plain terms why the true forward rate must be above $20\%$.

Because of compounding: the exact forward is about $20.2\%$. The ten-year investor's money compounds at $15\%$ throughout, so the base earning interest grows quickly. Someone who earns only $10\%$ for five years arrives at year 5 with a smaller pot, and must grow it faster than the simple average suggests to catch up: $(1+f)^5=\tfrac{1.15^{10}}{1.10^5}$, giving $f=\tfrac{1.15^2}{1.10}-1\approx20.2\%$.